The head light switch is shorting out ,or a bad connection
by design the DAY-TIME-RUNNING lights are not intended to be turned off by choice. When you start the 99 blazer with the parking brake on, the DTR lights are off. When you release the parking brake they come on. When you park again and apply the parking brake the DTR lights will stay on, unless you stop the engine and re-start the engine. Since the DTR lights are inter-wired to the regular headlights, they can not be defeated without affecting the regular headlights.
Check the fuse for the headlights. It is probably open.
Fuse for parking lights probably blown.
You can, but not while it's running. Before you turn your key, you have to set the parking brake, not all the way but just a bit and then start the Blazer and the lights should stay off.
Most likely a fuse or a bad ground.
Fuse, wiring, bulbs
Chevrolet Blazer parking brake can be adjusted with the adjustment screw on the parking brake pad. The parking brake can also be adjusted with the parking brake cable.
Try applying parking brake
On a 1998 Chevy Blazer, the parking brake switch is located on the top of the parking brake assembly. There are 3 bolts that hold the switch to the parking brake assembly.
You can only turn off the headlights by applying the emergency brake. When the emergency brake is on, the headlights will not light. Otherwise the best of my expererience is that they are on all the time. If you have remote start, put the emergency brake on before you leave the vehicle and when you remote start it, the headlights will not be on, but the driving lights (if switch is on) will be on as well as the parking lights.
In this order only. Turn engine off. Depress parking brake. Turn engine on. Lights will turn off. To turn lights back on relase parking brake.
Sounds like the switch is bad. You can buy a replacement switch at any automotive parts store and they are not that hard to install. Also check the fuse as sometimes the headlights have their own fuse while the parking/brake/back-up lights have their own.